This is just a general curiosity question:

In the standard textbook treatments of characteristic classes, and in particular the treatment of universal Pontrjagin classes, it's standard to consider $H^\ast(BSO,Z[1/2])$ (or $H^\ast(BSO(n),Z[1/2])$) in order to kill the 2-torsion. But I'm curious about that 2-torsion, since it should still give us some extra characteristic class-type information about real oriented bundles. If nothing else, it would give a class that's characteristic in the sense that it behaves the right way with respect to pullbacks (though I imagine it might be too much for these to be stable or have any kind of product formulas). So I suppose my questions are:

What's known about such classes?

Are they useful for anything?

Do these interact in any interesting way with the Stiefel-Whitney classes when everything is reduced mod 2?

Why are they usually ignored (or obliterated by coefficient change)?